For retrieval, instead of indexing \(N\)  document vectors, multi-vector retrieval pre-computes \(N \times \bar{m}\)  document token vectors where \(\bar{m}\)  is the average length of documents.
Then, it retrieves \(K\)  document token vectors for each query token vector with Maximum Inner-Product Search (MIPS), resulting in \(n \times K\)  candidate document tokens.
The retrieved tokens are used to trace back the original documents [1]}, often followed by a final refinement stage that scores the similarity \(\text{sim}(Q, D)\)  with all token representations of each document and the query [2]}.
We adopt the same practice of ColBERT in our experiments.
